How likely is a disproof of the conjecture that $2^q-1$ is squarefree whenever $q$ is prime?

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For the known Wieferich primes $1093$ and $3511$, we have the property that $$ord_2(p)=ord_2(p^2)$$ I have the following questions :




  • Is the property that the orders modulo $2$ of $p$ and $p^2$ coincide true for every Wieferich prime, or is it just a coincidence that it is true for the known Wieferich-primes ?

  • Can we estimate the chance, that a further Wieferich prime $p$ with, lets say, $20$ digits, has the property that $ord_2(p^2)$ is a prime ?



As far as I know, infinite many Wieferich primes are expected. If we would find a Wieferich prime $p$ such that $ord_2(p^2)$ is prime , we would have disproven that $2^q-1$ with prime exponent $q$ is always squarefree. I would like to know how likely this is.




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    For the known Wieferich primes $1093$ and $3511$, we have the property that $$ord_2(p)=ord_2(p^2)$$ I have the following questions :




    • Is the property that the orders modulo $2$ of $p$ and $p^2$ coincide true for every Wieferich prime, or is it just a coincidence that it is true for the known Wieferich-primes ?

    • Can we estimate the chance, that a further Wieferich prime $p$ with, lets say, $20$ digits, has the property that $ord_2(p^2)$ is a prime ?



    As far as I know, infinite many Wieferich primes are expected. If we would find a Wieferich prime $p$ such that $ord_2(p^2)$ is prime , we would have disproven that $2^q-1$ with prime exponent $q$ is always squarefree. I would like to know how likely this is.




    elementary-number-theory prime-numbers




    share|cite|improve this question






















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      For the known Wieferich primes $1093$ and $3511$, we have the property that $$ord_2(p)=ord_2(p^2)$$ I have the following questions :




      • Is the property that the orders modulo $2$ of $p$ and $p^2$ coincide true for every Wieferich prime, or is it just a coincidence that it is true for the known Wieferich-primes ?

      • Can we estimate the chance, that a further Wieferich prime $p$ with, lets say, $20$ digits, has the property that $ord_2(p^2)$ is a prime ?



      As far as I know, infinite many Wieferich primes are expected. If we would find a Wieferich prime $p$ such that $ord_2(p^2)$ is prime , we would have disproven that $2^q-1$ with prime exponent $q$ is always squarefree. I would like to know how likely this is.




      elementary-number-theory prime-numbers




      share|cite|improve this question












      For the known Wieferich primes $1093$ and $3511$, we have the property that $$ord_2(p)=ord_2(p^2)$$ I have the following questions :




      • Is the property that the orders modulo $2$ of $p$ and $p^2$ coincide true for every Wieferich prime, or is it just a coincidence that it is true for the known Wieferich-primes ?

      • Can we estimate the chance, that a further Wieferich prime $p$ with, lets say, $20$ digits, has the property that $ord_2(p^2)$ is a prime ?



      As far as I know, infinite many Wieferich primes are expected. If we would find a Wieferich prime $p$ such that $ord_2(p^2)$ is prime , we would have disproven that $2^q-1$ with prime exponent $q$ is always squarefree. I would like to know how likely this is.





      elementary-number-theory prime-numbers





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      asked Aug 18 at 7:46









      Peter

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